The invention described herein relates generally to flywheels, and more particularly to flywheels useful for energy storage as could be used in hybrid vehicle automotive power systems or in some stationary applications.
Prompted by recent national concern for energy conservation, there has been a substantial rebirth of interest in the technology of flywheels. Flywheels may be of use in energy storage systems and/or in land vehicle propulsion systems, inter alia. In particular, flywheels may be useful in power systems for hybrid automotive vehicles. Incorporating a flywheel in a battery powered vehicle may not only conserve energy but may also provide necessary acceleration and hill climbing performance.
Flywheels function as reservoirs of stored rotational kinetic energy. As rotational energy is withdrawn from a spinning flywheel, its angular speed decreases; as rotational energy is given to a flywheel, its angular speed increases. The kinetic energy of a flywheel as it rotates about its axis of symmetry is proportional to its moment of inertia multiplied by the square of its angular speed. The moment of inertia of a flywheel about its axis of symmetry depends upon the distribution of its mass. The angular speed of a flywheel cannot be increased without limit; at some angular speed, somewhere within the flywheel, the maximum allowable stress will be exceeded and the flywheel will permanently deform or, more probably, rupture and fly apart. Thus, other factors being equal, flywheels should be constructed of high strength material. Additionally, the energy per unit mass capable of being stored by a flywheel composed of an isotropic material would be maximized if the flywheel were so configured that, at rupture, its planar stresses were equal over its entire volume. This has been called a condition of two-dimensional hydrostatic stress which may be expressed by the rupture condition, throughout the entire flywheel, that: EQU s.sub.o =s.sub.r =s.sub.t
where
s.sub.o =the rupture stress of the flywheel material, a constant PA1 s.sub.r =the radial stress, and PA1 s.sub.t =the tangential stress. PA1 t=the thickness of the flywheel at radius r, PA1 t.sub.o =the thickness of the flywheel on the axis (r=0), PA1 .rho.=the material density of the flywheel, and PA1 .omega.=the angular speed of the flywheel at rupture. PA1 m=the mass of the infinite Stodola disk, PA1 E=the total kinetic energy in the Stodola flywheel at rupture, and PA1 .nu.=the ratio s.sub.o /.rho. or E/m, the energy density in the Stodola flywheel at rupture. PA1 t=the thickness of the body at radius r, PA1 t.sub.o =the thickness of the body on the axis (r=0), PA1 T=the thickness of the body where it is edge thickened, PA1 R=the maximum radial dimension of the body, and PA1 C=an arbitrary positive constant of dimensionality reciprocal length squared.
Such a configuration, which can only be approached in actual practice, was theoretically discovered for a hypothetical flywheel extending to infinite radius by engineers of the de Laval Company in Sweden circa 1900. This flywheel configuration, called a Stodola shape or disk because it was first published in a book by A. Stodola ("Steam and Gas Turbines", The McGraw-Hill Book Company, Inc., New York, N.Y., 1927), has a shape given by: EQU t=t.sub.o exp-(.rho..omega..sup.2 r.sup.2 /2s.sub.o),
where
The Stodola disk is also characterized by: EQU m.omega..sup.2 =2.pi.t.sub.o s.sub.o =2.pi.t.sub.o .rho..nu.=2.pi.t.sub.o .rho.E/m
where
Consequently, the theoretical maximum energy per unit mass that a flywheel composed of a planar isotropic material can store, and which could be stored by an infinite Stodola disk if one could be constructed, is given by the ratio of the stress at rupture to the density of the material of which the flywheel is composed.
Constructable flywheel shapes have been proposed that possess good energy density storage capability. These shapes are related to the Stodola shape. For example Call, in U.S. Pat. No. 3,496,799, discloses the use of a shaped energy storage flywheel having decreasing thickness as one moves toward the circumference of the flywheel. This flywheel has a truncated Stodola shape and is uncompensated for its failure to achieve planar stress equality in its peripheral regions.
The maximum strength to density ratio can be quite high for composites of high strength fibers (composed, for example, of various graphite compounds or glasses) embedded in various bonding materials such as epoxy resins. Because of this, composite flywheels were developed. Most of the composites of which flywheels have been constructed have all fibers lying parallel to one another. Such unidirectional composities are characterized by being very strong in the direction of the fibers, but, because of the relative weakness of the bonding materials, only about 1 or 2 percent as strong in the directions perpendicular to the fibers. Therefore, in order to properly utilize unidirectional composites in flywheel construction planar stress equality is undesirable and designs based on the Stodola shape are inappropriate. U.S. Pat. No. 4,028,962 to Nelson teaches shaped designs for flywheels composed of anisotropic materials that include non-monotonic thickness variation as radial distance increases. Also see Christensen and Wu, "Optimal Design of Anisotropic (Fiber-Reinforced) Flywheels", J. Composite Materials, 11, (1977) 395. Optimized anisotropic flywheel designs are unappealing because they tend toward either zero or infinite thickness along their axes of symmetry.
Recent work involving the application of high strength fiber composites in flywheel construction was presented by Satish V. Kulkarni at the "1979 Mechanical and Magnetic Energy Storage Contractors' Review Meeting", October 19-22, 1979 at Washington D.C. This material is published in, "Composite-Laminate Flywheel-Rotor Development Program", publication number UCRL-83554, by the Lawrence Livermore Laboratory of the University of California.